My analysis course states that a connected open set is path connected but I haven't been able to prove it. Here we consider a subset of the field of complex number but I suppose it works for a general topological space. Cheers
2026-04-02 16:26:27.1775147187
A connected open set is path connected
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Hint: Show that if $U$ is an open set in a normed space, then for each $x \in U$ the set of points that are path-connected to $x$ is open and closed (relative to $U$).